Categorical and contractible covers of polyhedra

Abstract

Let P be a compact connected polyhedron. A categorical (contractible) cover { X 1,…, X k } of P is a cover with the property that for each i=1,…, k, X i is null homotopic in P ( X i is contractible). The smallest integer k for which there is a categorical polyhedral cover of P with k elements is called the category of Lusternik-Schnirelmann of P, or simply the category of P, and is denoted by cat( P). Similarly, the smallest integer k for which there is a contractible polyhedral cover of P with k elements is called the geometric category of P and is denoted by gcat( P). Finally, the strong category of P, Cat( P), is the smallest integer k for which there is a polyhedron R with the homotopy type of P and such that k = gcat( R). The purpose of this paper is to develop some techniques which allow us to modify or to extend categorical or contractible covers of polyhedra, thus obtaining relations between the corresponding invariants.